This year, I’m teaching younger students than I’ve ever taught before. These guys are 11 and 12. They’re newer than iPods. They watched YouTube before they learned to read.

And so, instead of derivatives and arctangents, I find myself pondering more elemental ideas. Stuff I haven’t thought about in ages. Decimals. Perimeters. Rounding.

And most of all: Multiplication.

It’s dawning on me what a rich, complex idea multiplication is. It’s basic, but it isn’t easy. So many of the troubles that rattle and unsettle older students (factorization, square roots, compound fractions, etc.) can be traced back to a shaky foundation in this humble operation.

What’s so subtle about multiplication? Well, rather than just tell you, I’ll try to show you, by using a simple visualization of what it means to multiply.